A spherical ornament is placed into a cubic box for shipping so that its surface touches each of the faces of the box, as shown in the diagram below. The remaining volume that is not taken up by the spherical ornament is to be filled with a special packing material. If the radius of the spherical ornament is 4 centimeters, about how much space is left in the box to be filled with packing material?

243.92cm³

Step-by-step explanation:

We have two shapes given in this question. A spherical ornament and a cubic box

Step 1

Find the volume of the Sphere

Volume of the sphere is given as

4/3 πr³

In the question, the radius of the sphere is given as = 4 centimeters.

Therefore,

Volume of the sphere = 4/3 * π * 4³

Volume of the sphere = 268.08cm³

Step 2

We have to find the length of the side or the edge of the cube.

It is important note that: because the spherical ornament is insides the cubic box,

Hence, the diameter of the spherical ornament = length of the side (edge) of the cube.

In the question we are given the radius of the sphere = 4 cm

Diameter = 2 * radius = 2 * 4 cm = 8cm

Since, the diameter of the spherical ornament = length of the side (edge) of the cube,

The length of the side of the edge of the cube = 8cm

Step 3

We find the Volume of the cube

Volume of the cube = Length * Width * Height

Where the Length = Width = Height

Therefore, Voulme of the cube = 8cm * 8cm * 8cm

= 512cm³

Step 4

The fourth and final step is to find the space is left in the box to be filled with packing material.

The space left to be filled with packing material = Volume of the cube - Volume of the Spherical ornament

Volume of the cube = 512cm³ Volume of the Spherical ornament =

268.08cm³

Therefore, amount or Volume of space left for the packing material = 512cm³ - 268.08cm³

= 243.92cm³

Therefore, amount of space that is left in the box to be filled with packing material is 243.92cm³